Néron models. (English) Zbl 0705.14001

Bosch, Siegfried; Lütkebohmert, Werner; Raynaud, Michel

Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, 21. Berlin etc.: Springer-Verlag. x, 325 p. (1990).

Let \(S\) be a connected Dedekind scheme with field of rational functions \(K\) and let \(X_ K\) be a smooth and separated \(K\)-scheme of finite type. A Néron model of \(X_ K\) is a smooth separated \(S\)-scheme \(X\) of finite type with generic fibre \(X_ K\) satisfying the following universal property: for each smooth \(S\)-scheme \(Y\) and each \(K\)-morphism \(u_ K: Y_ K\to X_ K\) there exists a unique \(S\)-morphism \(u: Y\to X\) extending \(u_ k\).
This book is devoted to the construction of the Néron models and to the study of their properties. In particular, in the case of a relative curve \(X\to S\), the Néron model of the Jacobian \(J_ K\) of the generic fibre \(X_ K\) is studied and its relationship with the relative Picard functor is explained. The book contains also an ample exposition of the main tools and methods used so that, for example, chapter 8 and chapter 9 are a useful reference for questions related to the Picard functor and to its representability.

Reviewer: Luciana Picco Botta (Torino)

MSC:

14A15	Schemes and morphisms
14K30	Picard schemes, higher Jacobians
14L15	Group schemes
14-02	Research exposition (monographs, survey articles) pertaining to algebraic geometry
14C22	Picard groups

Keywords:

Néron model of the Jacobian; relative Picard functor